AIMS Mathematics, 2020, 5(4): 3809-3824. doi: 10.3934/math.2020247

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Existence, continuous dependence and finite time stability for Riemann-Liouville fractional differential equations with a constant delay

Department of Applied Mathematics and Modeling, University of Plovdiv “P. Hilendarski”, Plovdiv 4000, Bulgaria

Non-linear scalar Riemann-Liouville fractional differential equation with a constant delay is studied on a finite interval. An initial value problem is set up in appropriate way combining the idea of the initial time interval in ordinary differential equations with delays and the properties of Riemann-Liouville fractional derivatives. The mild solution of the studied initial value problem is defined. The existence, uniqueness, continuous dependence on the initial functions, finite time stability of the mild solutions are investigated.
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1. H. Zhang, R. Ye, S. Liu, et al. LMI-based approach to stability analysis for fractional-order neural networks with discrete and distributed delays, Int. J. Syst. Sci., 49 (2018), 537-545.    

2. W. Zhang, J. Cao, R. Wu, et al. Lag projective synchronization of fractional-order delayed chaotic systems, J. Franklin Institute, 356 (2019), 1522-1534.    

3. W. Zhang, H. Zhang, J. Cao, et al. Synchronization in uncertain fractional-order memristive complex-valued neural networks with multiple time delays, Neural Networks, 110 (2019), 186-198.    

4. P. Dorato, Short time stability in linear time-varying systems, Proc. IRE Int. Convention Record, 4 (1961), 83-87.

5. D. F. Luo, Z. G. Luo, Uniqueness and novel finite-time stability of solutions for a class of nonlinear fractional delay difference systems, Discr. Dynam, Nature Soc., 2018 (2018), 1-7.

6. G. C. Wu, D. Baleanu, S. D. Zeng, Finite-time stability of discrete fractional delay systems: Gronwall inequality and stability criterion, Commun. Nonl. Sci. Numer. Simul., 57 (2018), 299-308.    

7. V. N. Phat, N. T. Thanh, New criteria for finite-time stability of nonlinear fractional-order delay systems: A Gronwall inequality approach, Appl. Math. Lett., 83 (2018), 169-175.    

8. D. F. Luo, Z. G. Luo, Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses, Adv. Diff. Eq., 155, 2019.

9. D. Qian, C. Li, R. P. Agarwal, et al. Stability analysis of fractional differential system with Riemann-Liouville derivative, Math. Comput. Modell., 52 (2010), 862-874.    

10. M. Li, J. Wang, Finite time stability of fractional delay differential equations, Appl. Math. Lett., 64 (2017), 170-176.    

11. M. Li, J. R. Wang, Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations, Appl. Math. Comput., 324 (2018), 254-265.

12. M. Li, J. R. Wang, Finite time stability and relative controllability of Riemann-Liouville fractional delay differential equations, Math. Meth. Appl. Sci., 2019 (2019), 1-17.

13. K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2010.

14. I. Podlubny, Fractional Differential Equations, Academic Press: San Diego, 1999.

15. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

16. H. Ye, J. Gao, Y. Ding, A generalized Gronwall inequality and its application to a fracnal differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.    

17. R. Agarwal, S. Hristova, D. O'Regan, Explicit solutions of initial value problems for linear scalar Riemann-Liouville fractional differential equations with a constant delay, Mathematics, 8 (2020), 1-14.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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